section 4.5

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Section 4.5

Combining Probability and Counting Techniques

With some added content by D.R.S., University of Cordele






Example 4.34: Calculating Probability Using Combinations

A group of 12 tourists is visiting London. At one particular museum, a discounted admission is given to groups of at least ten. a. How many combinations of tourists can be made for

the museum visit so that the group receives the discounted rate?

b. Suppose that a group of the tourists does get the discount. What’s the probability that it was made up of 11 of the tourists?






Example 4.34: Calculating Probability Using Combinations

A group of 12 tourists is visiting London. At one particular museum, a discounted admission is given to groups of at least ten.

Analysis: They will get the museum discount if they bring how many people to the museum?

______ or ______ or ______ out of the group of 12choose to go to the museum.

Tourists group discount, part (a)

• How many ways to take 12 tourists 10 at a time? _______ = _______



• Use TI-84 nCr to find these values, not the primitive formula.

• So in all, how many ways to get “at least 10” for the museum? ______________________






Example 4.34: Calculating Probability Using Combinations (cont.)

b. Suppose that a group of the tourists does get the discount. What’s the probability that it was made up of 11 of the tourists?

Say it in conditional probability notation with words:P ( _________________ | _________________ )Calculate it:Numerator = _________Denominator = ________ or approximate decimal value: ____________






Example 4.35: Calculating Probability Using Permutations

Jack is setting a password on his computer. He is told that his password must contain at least three, but no more than five, characters. He may use either (the 26) letters or numbers (0–9). a. How many different possibilities are there for his

password if each character can only be used once? b. Suppose that Jack’s computer randomly sets his

password using all of the restrictions given above. What is the probability that this password would contain only the letters in his name?

Password Permutations, continued

• It’s a lot like the tourists.• The password can contain either _____ characters

or ____ characters or _____ characters.

• How many ways to take 36 characters ___ at a time?• How many ways to take 36 characters ___ at a time?• How many ways to take 36 characters ___ at a time?

• How many possible passwords in all? __________

Jack’s Password, continued.

b. Suppose that Jack’s computer randomly sets his password using all of the restrictions given above. What is the probability that this password would contain only the letters in his name?

First: How many passwords are there made up only of the characters “J”, “A”, “C”, and “K”? _____ = ______

Then: Form the probability:Numerator: How many J, A, C, K passwords: ______Denominator: How many possible passwords in all: _____Rounded decimal approximation: ______________

More with JACK

What if the question were “What is the probability that the password will contain all four letters of his name?”• What’s “new” here, besides the passwords we

counted in the previous problem?• How to count the number of passwords that meet

the requirements?• Will this be more probable, less probable, or the

same probability as before?• What is the probability now?






Example 4.36: Using Numbers of Combinations with the Fundamental Counting Principle

Tina is packing her suitcase to go on a weekend trip. She wants to pack 3 shirts, 2 pairs of pants, and 2 pairs of shoes. She has 9 shirts, 5 pairs of pants, and 4 pairs of shoes to choose from. How many ways can Tina pack her suitcase? (We will assume that everything matches.)

Suitcase, continued

• Permutations or Combinations?

• ______ shirts, taken _____ at a time: __________• ______ pants, taken _____ at a time: __________• ______ pair of shoes, taken _____ at a time: __________






Example 4.36: Using Numbers of Combinations with the Fundamental Counting Principle (cont.)

The final step in this problem is different than the final step in the last two examples, because of the word “and.” The Fundamental Counting Principle tells us we have to multiply, rather than add, the numbers of combinations together to get the final result. There are then

different ways that Tina can pack for the weekend.

84 10 6 5040






Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle

An elementary school principal is putting together a committee of 6 teachers to head up the spring festival. There are 8 first-grade, 9 second-grade, and 7 third grade teachers at the school. ‑a. In how many ways can the committee be formed? b. In how many ways can the committee be formed if

there must be 2 teachers chosen from each grade? c. Suppose the committee is chosen at random and

with no restrictions. What is the probability that 2 teachers from each grade are represented?







Solution In how many ways can the committee be formed?a. There are no stipulations as to what grades the

teachers are from and the order in which they are chosen does not matter, so we simply need to calculate the number of combinations by choosing 6 committee members from the 8 + 9 + 7 = 24 teachers. Calculate it here:

_____________ = ________________


b. How many ways if we get two from each grade?

Do you add (“or”) or multiply (“and”) ?

How many waysto fill thefirst-grade slots?

How many waysto fill thethird-grade slots?

How many waysto fill thesecond-grade slots?







c. P(a committee does have two from each grade)Compute the probability:

how many ways to get two from each gradehow many ways to form six overall any way

Should come up with 0.1573

Menu planning example

150 guests are coming.Select 5 appetizers, 4 entrees, 3 side dishes.Available 18 appetizers, 9 entrees, 12 side dishes.How many different menu plans are there?

Menu planning example

• Fundamental counting principle: • How many ways to choose appetizers, • times how many ways to choose entrees, • times how many ways to choose side dishes

• What about the 150 guests?

Reading list example

There are 10 novels, 8 non-fiction, and 6 self-help books on the shelf.

Choose 5 books to take on vacation.

How many different book choices?

Reading list example, modified

There are 10 novels, 8 non-fiction, and 6 self-help books on the shelf.

Choose 5 books to take on vacation, and at least one book must be a self-help book.

How many different book choices?


There are 10 novels, 8 non-fiction,18 books besides the self-help books, and 6 self-help books on the shelf.

Choose 5 books to take on vacation, and at least one book must be a self-help book.• How many ways to choose 1 self-help + 4 others• + How many ways to choose 2 self-help + 3 others• + How many ways to choose 3 self-help + 2 others• + How many ways to choose 4 self-help + 1 other• + How many ways to choose 5 self-help + 0 others


And… what if the order of the five books matters? Just calculate the number of ways to choose the five

books, to start with. Deal with “in order” in the post game show. Multiply the number of ways to choose the books

by 5!, since each of those ways has 5! possible orders.

Mixture of events of different natures

How many outcomes for this statistics triathlon?• Tossing a coin 5 times• Then drawing 2 cards without replacement• Then rolling a single six-sided die twice.

Mixing events of different natures

Fundamental Counting Principle, so multiply• The coin tosses are independent events• The card drawing is a combination• The die rolls are independent events.

Tossing a coin 10 times

… and the number of heads is between 5 and 8

Tossing a coin 10 times

… and the number of heads is between 5 and 8

• 10 coin tosses … the sample space has ______ outcomes.

• How many ways to get 5 heads? 10 C 5• How many ways to get 6 heads? 10 C 6• How many ways to get 7 heads? 10 C 7• How many ways to get 8 heads? 10 C 8

section 4.5

Documents

tourists group discount

combinations of tourists

success counts

museum discount

combinationsa group

particular museum

password permutations

discounted admission